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Abstract: Abstract We rigorously prove that the local conserved quantities in the one-dimensional Hubbard model are uniquely determined for each locality up to the freedom to add lower-order ones. From this, we can conclude that the local conserved quantities are exhausted by those obtained from the expansion of the transfer matrix. PubDate: 2024-06-04

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Abstract: Abstract The nonequilibrium Fokker–Planck dynamics with a non-conservative drift field, in dimension \(N\ge 2\) , can be related with the non-Hermitian quantum mechanics in a real scalar potential V and in a purely imaginary vector potential -iA of real amplitude A (Mazzolo and Monthus in Phys Rev E 107:014101, 2023). Since Fokker–Planck probability density functions may be obtained by means of Feynman’s path integrals, the previous observation points towards a general issue of “magnetically affine” propagators, possibly of quantum origin, in real and Euclidean time. In below we shall follow the \(N=3\) “magnetic thread”, within which one may keep under a computational control formally and conceptually different implementations of magnetism (or surrogate magnetism) in the dynamics of diffusion processes. We shall focus on interrelations (with due precaution to varied, not evidently compatible, notational conventions) of: (i) the pertinent non-conservatively drifted diffusions, (ii) the classic Brownian motion of charged particles in the (electro)magnetic field, (iii) diffusion processes arising within so-called Euclidean quantum mechanics (which from the outset employs non-Hermitian “magnetic” Hamiltonians), (iv) limitations of the usefulness of the Euclidean map \(\exp (-itH_{quant}) \rightarrow \exp (-tH_{Eucl})\) , regarding the probabilistic significance of inferred (path) integral kernels in the description of diffusion processes. PubDate: 2024-05-30

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Abstract: Abstract Unlike the single species gases, the transport coefficients such as Fick, Soret, Dufour coefficients arise in the hydrodynamic limit of multi-species gas mixtures. To the best of the authors’ knowledge, no multi-component relaxational models is reported that produces all these values correctly. In this paper, we establish the existence of unique stationary mild solutions to the BGK models for gas mixtures which produces the correct Fick coefficients in the Navier–Stokes limit for inert gases (Brull in Eur J Mech B 33:74–86, 2012), and for reactive gases (Brull and Schneider in Commun Math Sci 12(7):1199–1223, 2014) in a unified manner. PubDate: 2024-05-30

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Abstract: Abstract We present a statistical mechanical theory of multi-component fluids, where we consider the correlation functions of the number densities and the energy density in the grand canonical ensemble. In terms of their space integrals we express the partial volumes \({{\bar{v}}}_i\) , the partial enthalpies \({{\bar{H}}}_i\) , and other thermodynamic derivatives. These \({{\bar{v}}}_i\) and \({{\bar{H}}}_i\) assume simple forms for binary mixtures and for ternary mixtures with a dilute solute. They are then related to the space-dependent thermal fluctuations of the temperature and the pressure. The space averages of these fluctuations are those introduced by Landau and Lifshits in the isothermal-isobaric (T-p) ensemble. We also give expressions for the long-range (nonlocal) correlations in the canonical and T-p ensembles, which are inversely proportional to the system volume. For a mixture solvent, we examine the solvent-induced solute–solute attraction and the osmotic enthalpy changes due to the solute doping using the correlation function integrals. PubDate: 2024-05-30

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Abstract: Abstract In this work, we develop a mathematical framework to model a quantum system whose time evolution may depend on the state of a randomly changing environment that evolves according to a Markovian process. When the environment changes its state, three possible things may occur: the quantum system starts evolving according to a new Hamiltonian, it may suffer an instantaneous perturbation that changes its state or both things may happen simultaneously. We consider the case of quantum systems with finite dimensional Hilbert state space, in which case the observables are described by Hermitian matrices. We show how to average over the environment to predict the expected value of the density matrix with which one can compute the expected values of the observables of interest. PubDate: 2024-05-30

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Abstract: Abstract We consider the relaxation of an initial non-equilibrium state in a one-dimensional fluid of hard rods. Since it is an interacting integrable system, we expect it to reach the Generalized Gibbs Ensemble (GGE) at long times for generic initial conditions. Here we show that there exist initial conditions for which the system does not reach GGE even at very long times and in the thermodynamic limit. In particular, we consider an initial condition of uniformly distributed hard-rods in a box with the left half having particles with a singular velocity distribution (all moving with unit velocity) and the right half particles in thermal equilibrium. We find that the density profile for the singular component does not spread to the full extent of the box and keeps moving with a fixed effective speed at long times. We show that such density profiles can be well described by the solution of the Euler equations almost everywhere except at the location of the shocks, where we observe slight discrepancies due to dissipation arising from the initial fluctuations of the thermal background. To demonstrate this effect of dissipation analytically, we consider a second initial condition with a single particle at the origin with unit velocity in a thermal background. We find that the probability distribution of the position of the unit velocity quasi-particle has diffusive spreading which can be understood from the solution of the Navier–Stokes (NS) equation of the hard rods. Finally, we consider an initial condition with a spread in velocity distribution for which we show convergence to GGE. Our conclusions are based on molecular dynamics simulations supported by analytical arguments. PubDate: 2024-05-30

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Abstract: Abstract Local quantum Bernoulli noises (LQBNs) is a family of local annihilation operators and local creation operators acting on Bernoulli functions. In this paper, based on LQBNs, we define the local quantum joint entropy and local quantum coherence information. And we find local quantum entropy has a surprising property, which is not satisfies the additivity for tensor product state. Furthermore, we give some properties of local quantum coherent information based on LQBNs. PubDate: 2024-05-28

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Abstract: Abstract We consider general classes of gradient models on regular trees with spin values in a countable Abelian group S such as \({\mathbb {Z}}\) or \({\mathbb {Z}}_q\) . This includes unbounded spin models like the p-SOS model and finite-alphabet clock models. Under a strong coupling (low temperature) condition on the interaction, we prove the existence of families of distinct homogeneous tree-indexed Markov chain Gibbs states \(\mu _A\) whose single-site marginals concentrate on a given finite subset \(A\subset S\) of spin values. The existence of such states is a new and robust phenomenon which is of particular relevance for infinite spin models. These states are extremal in the set of homogeneous Gibbs states, and in particular cannot be decomposed into homogeneous Markov-chain Gibbs states with a single-valued concentration center. Whether they are also extremal in the set of all Gibbs states remains an open, challenging question. As a further application of the method we obtain the existence of new types of gradient Gibbs states with \({\mathbb {Z}}\) -valued spins, whose single-site marginals do not localize, but whose correlation structure depends on the finite set A, where we provide explicit expressions for the correlation between the height-increments along disjoint edges. PubDate: 2024-05-27

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Abstract: Abstract We solve the Random Euclidean Matching problem with exponent 2 for the Gaussian distribution defined on the plane. Previous works by Ledoux and Talagrand determined the leading behavior of the average cost up to a multiplicative constant. We explicitly determine the constant, showing that the average cost is proportional to \((\log \, N)^2,\) where N is the number of points. Our approach relies on a geometric decomposition allowing an explicit computation of the constant. Our results illustrate the potential for exact solutions of random matching problems for many distributions defined on unbounded domains on the plane. PubDate: 2024-05-16

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Abstract: Abstract We study the passive transport of a scalar field by a spatially smooth but white-in-time incompressible Gaussian random velocity field on \(\mathbb {R}^d\) . If the velocity field u is homogeneous, isotropic, and statistically self-similar, we derive an exact formula which captures non-diffusive mixing. For zero diffusivity, the formula takes the shape of \(\mathbb {E}\ \Vert \theta _t \Vert _{\dot{H}^{-s}}^2 = \textrm{e}^{-\lambda _{d,s} t} \Vert \theta _0 \Vert _{\dot{H}^{-s}}^2\) with any \(s\in (0,d/2)\) and \(\frac{\lambda _{d,s}}{D_1}:= s(\frac{\lambda _{1}}{D_1}-2s)\) where \(\lambda _1/D_1 = d\) is the top Lyapunov exponent associated to the random Lagrangian flow generated by u and \( D_1\) is small-scale shear rate of the velocity. Moreover, the mixing is shown to hold uniformly in diffusivity. PubDate: 2024-05-15

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Abstract: Abstract The Lillo–Mike–Farmer (LMF) model is an established econophysics model describing the order-splitting behaviour of institutional investors in financial markets. In the original article (Lillo et al. in Phys Rev E 71:066122, 2005), LMF assumed the homogeneity of the traders’ order-splitting strategy and derived a power-law asymptotic solution to the order-sign autocorrelation function (ACF) based on several heuristic reasonings. This report proposes a generalised LMF model by incorporating the heterogeneity of traders’ order-splitting behaviour that is exactly solved without heuristics. We find that the power-law exponent in the order-sign ACF is robust for arbitrary heterogeneous order-submission probability distributions. On the other hand, the prefactor in the ACF is very sensitive to heterogeneity in trading strategies and is shown to be systematically underestimated in the original homogeneous LMF model. Our work highlights that predicting the ACF prefactor is more challenging than the ACF exponent because many microscopic details (complex ingredients in actual data analyses) start to matter. PubDate: 2024-05-07

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Abstract: Abstract Unlike the case for classical particles, the literature on BGK type models for relativistic gas mixture is extremely limited. There are a few results in which such relativistic BGK models for gas mixture are employed to compute transport coefficients. However, to the best knowledge of authors, relativistic BGK models for gas mixtures with complete presentation of the relaxation operators are missing in the literature. In this paper, we fill this gap by suggesting a BGK model for relativistic gas mixtures for which the existence of each equilibrium coefficients in the relaxation operator is rigorously guaranteed in a way that all the essential physical properties are satisfied such as the conservation laws, the H-theorem, the capturing of the correct equilibrium state, the indifferentiability principle, and the recovery of the classical BGK model in the Newtonian limit. PubDate: 2024-05-07

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Abstract: Abstract We consider a random field \(\phi ({\textbf{r}})\) in d dimensions which is largely concentrated around small ‘hotspots’, with ‘weights’, \(w_i\) . These weights may have a very broad distribution, such that their mean does not exist, or is dominated by unusually large values, thus not being a useful estimate. In such cases, the median \({\overline{W}}\) of the total weight W in a region of size R is an informative characterisation of the weights. We define the function F by \(\ln {\overline{W}}=F(\ln R)\) . If \(F'(x)>d\) , the distribution of hotspots is dominated by the largest weights. In the case where \(F'(x)-d\) approaches a constant positive value when \(R\rightarrow \infty \) , the hotspots distribution has a type of scale-invariance which is different from that of fractal sets, and which we term ultradimensional. The form of the function F(x) is determined for a model of diffusion in a random potential. PubDate: 2024-05-07

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Abstract: Abstract We analyse the motion of one particle in a polymer chain. For this purpose, we use the framework of the exact (non-stationary) generalized Langevin equation that can be derived from first principles via the projection-operator method. Our focus lies on determining memory kernels from either exact expressions for autocorrelation functions or from simulation data. We increase the complexity of the underlying system starting out from one-dimensional harmonic chains and ending with a polymer driven through a polymer melt. Here, the displacement or the velocity of an individual particle in the chain serves as the observable. The central result is that the time-window in which the memory kernels show structure before they rapidly decay decreases with increasing complexity of the system. PubDate: 2024-05-04

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Abstract: Abstract Adding activity or driving to a thermal system may modify its phase diagram and response functions. We study that effect for a Curie–Weiss model where the thermal bath switches rapidly between two temperatures. The critical temperature moves with the nonequilibrium driving, opening up a new region of stability for the paramagnetic phase (zero magnetization) at low temperatures. Furthermore, phase coexistence between the paramagnetic and ferromagnetic phases becomes possible at low temperatures. Following the excess heat formalism, we calculate the nonequilibrium thermal response and study its behaviour near phase transitions. Where the specific heat at the critical point makes a finite jump in equilibrium (discontinuity), it diverges once we add the second thermal bath. Finally, (also) the nonequilibrium specific heat goes to zero exponentially fast with vanishing temperature, realizing an extended Third Law. PubDate: 2024-05-04

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Abstract: Abstract We study the competition between Haar-random unitary dynamics and measurements for unstructured systems of qubits. For projective measurements, we derive various properties of the statistical ensemble of Kraus operators analytically, including the purification time and the distribution of Born probabilities. The latter generalizes the Porter–Thomas distribution for random unitary circuits to the monitored setting and is log-normal at long times. We also consider weak measurements that interpolate between identity quantum channels and projective measurements. In this setting, we derive an exactly solvable Fokker–Planck equation for the joint distribution of singular values of Kraus operators, analogous to the Dorokhov–Mello–Pereyra–Kumar (DMPK) equation modelling disordered quantum wires. We expect that the statistical properties of Kraus operators we have established for these simple systems will serve as a model for the entangling phase of monitored quantum systems more generally. PubDate: 2024-05-03

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Abstract: Abstract Granular gases demand models capable of capturing their distinct characteristics. The widely employed inelastic hard-sphere model (IHSM) introduces complexities that are compounded when incorporating realistic features like surface roughness and rotational degrees of freedom, resulting in the more intricate inelastic rough hard-sphere model (IRHSM). This paper focuses on the inelastic rough Maxwell model (IRMM), presenting a more tractable alternative to the IRHSM and enabling exact solutions. Building on the foundation of the inelastic Maxwell model (IMM) applied to granular gases, the IRMM extends the mathematical representation to encompass surface roughness and rotational degrees of freedom. The primary objective is to provide exact expressions for the Navier–Stokes–Fourier transport coefficients within the IRMM, including the shear and bulk viscosities, the thermal and diffusive heat conductivities, and the cooling-rate transport coefficient. In contrast to earlier approximations in the IRHSM, our study unveils inherent couplings, such as shear viscosity to spin viscosity and heat conductivities to counterparts associated with a torque-vorticity vector. These exact findings provide valuable insights into refining the Sonine approximation applied to the IRHSM, contributing to a deeper understanding of the transport properties in granular gases with realistic features. PubDate: 2024-04-27

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Abstract: Abstract We establish a connection between the relative Classical entropy and the relative Fermi–Dirac entropy, allowing to transpose, in the context of the Boltzmann or Landau equation, any entropy–entropy production inequality from one case to the other; therefore providing entropy–entropy production inequalities for the Boltzmann–Fermi–Dirac operator, similar to the ones of the Classical Boltzmann operator. We also provide a generalized version of the Csiszár–Kullback–Pinsker inequality to weighted \(L^p\) norms, \(1 \le p \le 2\) and a wide class of entropies. PubDate: 2024-04-27

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Abstract: Abstract This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker–Planck equations with non-quadratic confinement potentials in whole space. We extend previous approaches by analyzing Lyapunov functionals with non-constant weight matrices in the dissipation functional (a generalized Fisher information). We establish exponential convergence in a weighted \(H^1\) -norm with rates that become sharp in the case of quadratic potentials. In the defective case for quadratic potentials, i.e. when the drift matrix has non-trivial Jordan blocks, the weighted \(L^2\) -distance between a Fokker–Planck-solution and the steady state has always a sharp decay estimate of the order \(\mathcal O\big ( (1+t)e^{-t\nu /2}\big )\) , with \(\nu \) the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker–Planck equations (from a weighted \(L^2\) -space to a weighted \(H^1\) -space). PubDate: 2024-04-27

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Abstract: Abstract A half-space problem of a linear kinetic equation for gas molecules physisorbed close to a solid surface, relevant to a kinetic model of gas–surface interaction and derived by Aoki et al. (Phys. Rev. E 106:035306, 2022), is considered. The equation contains a confinement potential in the vicinity of the solid surface and an interaction term between gas molecules and phonons. It is proved that a unique solution exists when the incoming molecular flux is specified at infinity. This validates the natural observation that the half-space problem serves as the boundary condition for the Boltzmann equation. It is also proved that the sequence of approximate solutions used for the existence proof converges exponentially fast. In addition, numerical results showing the details of the solution to the half-space problem are presented. PubDate: 2024-04-27